Let (M, ω) be a geometrically bounded symplectic manifold, N ⊆ M a closed, regular (i.e., "fibering") coisotropic submanifold, and ϕ : M → M a Hamiltonian diffeomorphism. The main result of this article is that the number of leaf-wise fixed points of ϕ is bounded below by the sum of the Z 2 -Betti numbers of N , provided that the Hofer distance between ϕ and the identity is small enough and the pair (N, ϕ) is non-degenerate. The bound is optimal if there exists a Z 2 -perfect Morse function on N . A version of the Arnol'd-Givental conjecture for coisotropic submanifolds is also discussed. As an application, I prove a presymplectic non-embedding result.

The (local) invariant symplectic action functional A is associated to a Hamiltonian action of a compact connected Lie group G on a symplectic manifold (M, ω), endowed with a Ginvariant Riemannian metric •, • M . It is defined on the set of pairs of loops (x, ξ) : S 1 → M × Lie G for which x satisfies some admissibility condition. I prove a sharp isoperimetric inequality for A if •, • M is induced by some ω-compatible and G-invariant almost complex structure J, and, as an application, an optimal result about the decay at ∞ of symplectic vortices on the half-cylinder [0, ∞) × S 1 .

We prove a coisotropic intersection result and deduce the following: (a) Lower bounds on the displacement energy of a subset of a symplectic manifold, in particular a sharp stable energy-Gromov-width inequality. (b) A stable non-squeezing result for neighborhoods of products of unit spheres. (c) Existence of a "badly squeezable" set in R 2n of Hausdorff dimension at most d, for every n ≥ 2 and d ≥ n. (d) Existence of a stably exotic symplectic form on R 2n , for every n ≥ 2. (e) Non-triviality of a new capacity, which is based on the minimal action of a regular coisotropic submanifold of dimension d.

Consider a closed coisotropic submanifold N of a symplectic manifold (M, ω) and a Hamiltonian diffeomorphism ϕ on M . The main result of this article states that ϕ has at least the cup-length of N many leafwise fixed points w.r.t. N , provided that it is the time-1-map of a global Hamiltonian flow whose restriction to N stays C 0 -close to the inclusion N → M . If (ϕ, N ) is suitably nondegenerate then the number of these points is bounded below by the sum of the Betti-numbers of N . The nondegeneracy condition is generically satisfied.This appears to be the first leafwise fixed point result in which neither ϕ N is assumed to be C 1 -close to the inclusion N → M , nor N to be of contact type or regular (i.e., "fibering"). It is optimal in the sense that the C 0condition on ϕ cannot be replaced by the assumption that ϕ is Hofer-small.

We show that the spherical capacity is discontinuous on a smooth family of ellipsoidal shells. Moreover, we prove that the shell capacity is discontinuous on a family of open sets with smooth connected boundaries.Comment: We include generalizations to higher dimensions due to the unknown referee and Janko Latschev. We add examples of open sets with connected boundary on which the shell capacity is not continuous. 3rd and 4th version: minor changes, to appear in J. Fixed Point Theory App